Floating immittance emulator

ABSTRACT

The floating immittance emulator is presented in four embodiments in which four new topologies for emulating floating immittance functions are detailed. Each circuit uses three current-feedback operational-amplifiers (CFOAs) and three passive elements. The present topologies can emulate lossless and lossy floating inductances; capacitance, resistance, and inductance multipliers; and frequency-dependent positive and negative resistances.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a divisional of U.S. patent application Ser. No.14/951,467, filed Nov. 24, 2015, now pending.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to emulator circuits, particularly tofloating immittance emulator circuits that use three current-feedbackoperational amplifiers (CFOAs).

2. Description of the Related Art

Over the years researchers have reported several floating inductancesimulators using a wide range of active elements. This is attributed toits importance in designing many analog signal processing circuits, suchas impedance matching circuits, low frequency filters, and oscillators,where relatively large values of inductance that cannot be fabricated onthe chip are required. Of particular interest are realizations based onthe use of the CFOA as an active element. This is attributed to theunique characteristics of the CFOA, such as the relatively wideroperating bandwidth (there is no gain-bandwidth limitation), itsrelatively high slew rate, and its commercial availability. Obviously,the use of the minimum number of CFOAs is preferable, as it implies lesspower consumption and less area on the chip.

Thus, a floating immittance emulator solving the aforementioned problemsis desired.

SUMMARY OF THE INVENTION

The floating immittance emulator is presented in various circuits foremulating immittance (impedance or admittance [ratio of current tovoltage]; immittance is a term embracing both). Each circuit uses threecurrent-feedback operational-amplifiers (CFOAs), and passive elements.The present topologies can emulate lossless and lossy floatinginductances, and capacitance, resistance and inductance multipliers, inaddition to frequency-dependent positive and negative resistances. Thefunctionality of the present circuits is verified using Advanced DesignSystem (ADS) software and the AD844 CFOA. The simulation results are inexcellent agreement with the theoretical calculations.

These and other features of the present invention will become readilyapparent upon further review of the following specification anddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a schematic diagram of a first embodiment of a floatingimmittance emulator according to the present invention.

FIG. 1B is a schematic diagram of a second embodiment of a floatingimmittance emulator according to the present invention.

FIG. 1C is a schematic diagram of a third embodiment of a floatingimmittance emulator according to the present invention.

FIG. 1D is a schematic diagram of a fourth embodiment of a floatingimmittance emulator according to the present invention.

FIG. 2 is a block diagram of a test circuit for the floating immittanceemulators of FIGS. 1A-1D.

FIG. 3 is a plot showing the transfer function of a bandpass filterobtained using the inductance emulator of FIG. 1A

FIG. 4 is a plot showing the transfer function of a bandpass filterobtained using the inductance emulator of FIG. 1B

FIG. 5 is a plot showing waveforms of the current through and thevoltage across an inductance emulated using FIG. 1C

FIG. 6 is a plot showing the transfer function of a bandpass filterobtained using the immittance emulator of FIG. 1C

FIG. 7 is a plot showing the transfer function of another bandpassfilter obtained using the immittance emulator of FIG. 1C

FIG. 8 is a plot showing waveforms of the current through and thevoltage across a capacitance emulated using the immittance emulator ofFIG. 1C

FIG. 9 is a plot showing variation of the capacitance obtained using acapacitance multiplier emulated using the immittance emulator of FIG. 1C

FIG. 10 is a plot showing variation of the resistance obtained using aresistance multiplier emulated using the immittance emulator of FIG. 1C.

FIG. 12 is a plot showing waveforms of the current through and thevoltage

FIG. 11 is a plot showing variation of the inductance obtained using theinductance multiplier emulated using the immittance emulator of FIG. 1C.across a frequency-dependent negative-resistance emulated using theimmittance emulator of FIG. 1C

FIG. 13 is a plot showing waveforms of the current through and the inputvoltage of the circuit built using the negative inductance emulatedusing the immittance emulator of FIG. 1D

Similar reference characters denote corresponding features consistentlythroughout the attached drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

As shown in FIGS. 1A, 1B, 1C and 1D, the three-CFOA-based floatingimmittance emulator includes inductance simulators using three CFOAs.Regarding references to the y, x, z, and w-terminals of the CFOAs, itwill be understood that as used herein, the y- and x-terminals are inputterminals of a CFOA building block and z- and w-output terminals of theCFOA comprise the z-output terminal (the slewing node of the AnalogDevices AD844) and w-output terminal, respectively, of the CFOA.

The immittance emulator circuit 100 a shown in FIG. 1A includes a firstCFOA 112 a having first y-, x-, z-, and w-terminals, first z-terminalbeing connected to ground. A second CFOA 112 b having second y-, x-, z-,and w-terminals, the second z-terminal being connected to the secondy-terminal. A third CFOA 112 c having third y-, x-, z-, and w-terminals,the third z-terminal being connected to the second x-terminal of thesecond CFOA. A first impedance 101 has a first lead connected to thefirst y-terminal of the first CFOA 112 a and a second lead connected tothe third x-terminal of the third CFOA 112 c. A second impedance 102having a first lead connected to the first x-terminal of first CFOA 112a and a second lead connected to the third y-terminal of third CFOA 112c. A third impedance 103 having a first lead connected to the thirdy-terminal of third CFOA 112 c and a second lead connected to the thirdw-terminal of third CFOA 112 c. This circuit performs immittanceemulation between a voltage v₁ applied to the first y-terminal of thefirst CFOA 112 a and a voltage v₂ applied to the second y-terminal ofthe second CFOA 112 b.

The immittance emulator circuit 100 b shown in FIG. 1B includes a firstCFOA 112 a having first y-, x-, z-, and w-terminals. A second CFOA 112 bhaving second y-, x-, z-, and w-terminals. A third CFOA 112 c havingthird y-, x-, z-, and w-terminals, the third z-terminal being connectedto the first x-terminal of the first CFOA 112 a. A first impedance 101has a first lead connected to the first y-terminal of first CFOA 112 aand a second lead connected to the third x-terminal of third CFOA 112 c.A second impedance 102 has a first lead connected to the thirdw-terminal of third CFOA 112 c and a second lead connected to the thirdy-terminal of third CFOA 112 c. A third impedance 103 has a first leadconnected to the third y-terminal of third CFOA 112 c and a second leadconnected to the second x-terminal of second CFOA 112 b. The secondz-terminal of the second CFOA 112 b is connected to ground. The secondy-terminal of the second CFOA 112 b is connected to first z-terminal ofCFOA 112 a. This circuit performs immittance emulation between a voltagev₁ applied to the first y-terminal of the first CFOA 112 a and a voltagev₂ applied to the second y-terminal of the second CFOA 112 b.

The immittance emulator circuit 100 c shown in FIG. 1C includes a firstCFOA 112 a having first y-, x-, z-, and w-terminals. A second CFOA 112 bhas second y-, x-, z-, and w-terminals. A third CFOA 112 c has third y-,x-, z-, and w-terminals, the third z-terminal being connected to thefirst x-terminal of the first CFOA 112 a, the third y-terminal beingconnected to the second w-terminal of CFOA 112 b. A first impedance 101has a first lead connected to the first y-terminal of the first CFOA 112a and a second lead connected to the third x-terminal of the third CFOA112 c. A second impedance 102 has a first lead connected to the thirdw-terminal of the third CFOA 112 c and a second lead connected to thesecond z-terminal of the second CFOA 112 b. A third impedance 103 has afirst lead connected to the third w-terminal of the third CFOA 112 c anda second lead connected to the second x-terminal of the second CFOA 112b. The second y-terminal of the second CFOA 112 b is connected to thefirst z-terminal of CFOA 112 a. This circuit performs immittanceemulation between a voltage v₁ applied to the first y-terminal of thefirst CFOA 112 a and a voltage v₂ applied to the second y-terminal ofthe second CFOA 112 b.

The immittance emulator circuit 100 d shown in FIG. 1D includes a firstCFOA 112 a having first y-, x-, z-, and w-terminals. A second CFOA 112 bhas second y-, x-, z-, and w-terminals. A third CFOA 112 c has third y-,x-, z-, and w-terminals, the third x-terminal being connected to thefirst x-terminal of the first CFOA 112 a and the second x-terminal ofthe second CFOA 112 b. A first impedance 101 has a first lead connectedto the first y-terminal of the first CFOA 112 a and a second leadconnected to the first z-terminal of the first CFOA 112 a and the secondz-terminal of the second CFOA 112 b. A second impedance 102 has a firstlead connected to the first w-terminal of the first CFOA 112 a and asecond lead connected to the third y-terminal of the third CFOA 112 c. Athird impedance 103 has a first lead connected to the third y-terminalof the third CFOA 112 c and a second lead connected to the thirdw-terminal of the third CFOA 112 c. The second y-terminal of the secondCFOA 112 b is connected to first y-terminal of CFOA 112 a. The secondw-terminal of the second CFOA 112 b is connected to the first w-terminalof CFOA 112 a. This circuit performs immittance emulation between avoltage v₁ applied to the first y-terminal of the first CFOA 112 a and avoltage v₂ applied to the third z-terminal of the third CFOA 112 c.

Assuming that the CFOAs are characterized by v_(y)=v_(x), i_(z)=i_(x),v_(w)=v_(z), i_(y)=0, routine analysis shows that the input impedancebetween terminals 1 and 2 of FIGS. 1A-1D is given by

$\begin{matrix}{{Z_{in} = {\frac{V_{1} - V_{2}}{I_{in}} = {Z_{1} + \frac{Z_{1}Z_{3}}{Z_{2}}}}},} & (1)\end{matrix}$for the circuits of FIGS. 1A and 1B,

$\begin{matrix}{{Z_{in} = {\frac{V_{1} - V_{2}}{I_{in}} = \frac{Z_{1}Z_{3}}{Z_{2}}}},} & (2)\end{matrix}$for the circuit of FIG. 1C, and

$\begin{matrix}{{Z_{in} = {\frac{V_{1} - V_{2}}{I_{in}} = {- \frac{Z_{1}Z_{3}}{Z_{2}}}}},} & (3)\end{matrix}$for the circuit of FIG. 1D.

Thus, with

${Z_{1} = R_{1}},{Z_{2} = \frac{1}{{sC}_{2}}},$Z₃=R₃, then the circuits of FIGS. 1A and 1B can simulate a lossyinductance with a series connected resistance with R_(eq)=R₁,L_(eq)=C₂R₁R₃, the circuit of FIG. 1C can simulate a positive losslessinductance with L_(eq)=C₂R₁R₃, and the circuit of FIG. 1D can simulate alossless negative inductance with L_(eq)=−C₂R₁R₃. Moreover, with

${Z_{1} = \frac{1}{{sC}_{1}}},{Z_{3} = \frac{1}{{sC}_{3}}}$and, Z₂=R₂ the circuit of FIG. 1C can simulate a positive frequencydependent resistance given by:

$R_{eq} = \frac{1}{\omega^{2}C_{1}C_{3}R_{2}}$and the circuit of FIG. 1D can simulate a negative frequency dependentresistance given by:

$R_{eq} = {\frac{- 1}{\omega^{2}C_{1}C_{3}R_{2}}.}$Furthermore, with Z₁=R₁, Z₃=R₃ and Z₂=R₂, the circuit of FIG. 1Crealizes a positive resistance multiplier given by:

$R_{eq} = \frac{R_{1}R_{3}}{R_{2}}$and the circuit of FIG. 1D can simulate a negative resistance multipliergiven by:

$R_{eq} = {- {\frac{R_{1}R_{3}}{R_{2}}.}}$Finally, with Z₁=R₁, Z₂=R₂ and

${Z_{3} = \frac{1}{{sC}_{3}}},$the circuit of FIG. 1C can simulate a positive capacitance multiplierwith:

$C_{eq} = \frac{R_{2}C_{3}}{R_{1}}$and the circuit of FIG. 1D can simulate a negative capacitancemultiplier with:

$C_{eq} = {- {\frac{R_{2}C_{3}}{R_{1}}.}}$

The proposed circuits of FIGS. 1A, 1B, 1C and 1D were simulated usingthe CFOA specified as Analog Devices AD844 with DC supplyvoltages=±5.0V. The proposed lossy floating positive inductor obtainablefrom FIG. 1A was used in the test bench circuit of FIG. 2 with Z_(i)formed of a capacitance C_(i)=1.0 μF and the resistance R_(o)=2.5 kΩ,and the values of the components in FIG. 1A selected as R₁=1.0 kΩ,C₂=1.0 μF and R₃ as a variable resistor in the range 1.0 kΩ-50.0 kΩ. Theoutput voltage across the resistance R_(o) was monitored, and theresults obtained are shown in FIG. 3. Inspection of FIG. 3 shows thatthe circuit behaves as a bandpass filter with variable Q and centerfrequency. Calculations using the simulation results show that theemulated inductance has a loss equivalent to 1.07 kΩ, which agrees wellwith the theoretical calculation of 1.0 kΩ loss. Moreover, inspection ofFIG. 3 shows that the center frequency varies between 21.0 Hz and 151.0Hz. This is in excellent agreement with the calculations showing thatthe center frequency changes from 22.5 to 159.0 Hz. This confirms thatthe circuit of FIG. 1A emulates a lossy positive floating inductance.

The proposed lossy floating positive inductor obtainable from FIG. 1Bwas used in the test bench circuit 200 of FIG. 2 with external impedanceZ_(i) formed of a capacitance C_(i)=1.0 μF and the resistance R_(o)=2.5kΩ, and the values of the components in FIG. 1B selected as R₁=1.0 kΩ,C₂=1.0 μF and R₃ variable in the range 1.0 kΩ-50.0 kΩ. The outputvoltage across the resistance R_(o) was monitored, and the resultsobtained are shown in FIG. 4. Inspection of FIG. 4 shows that thecircuit behaves as a bandpass filter with variable Q and centerfrequency. This confirms that the circuit of FIG. 1A simulates a lossypositive floating inductance. The simulation results show that thecenter frequency changes from 22.5 Hz to 159.0 Hz. This is in excellentagreement with the calculations showing that the center frequencychanges between 21.0 Hz and 151.0 Hz.

The proposed lossless floating positive inductor obtainable from FIG. 1Cwas used in the test bench circuit of FIG. 2 with Z_(i) formed of aresistance R_(i)=314Ω and the resistance R_(o)=314Ω, and the values ofthe components in FIG. 1C selected as R₁=1.0 kΩ, C₂=50.0 nF and R₃=1.0kΩ. The current through the emulated inductance and the voltage acrossit were monitored, and the results obtained are shown in plot 500 ofFIG. 5. Inspection of FIG. 5 shows that the current through the emulatedinductance lags by 90° behind the voltage across it. The proposedlossless floating positive inductor obtainable from FIG. 1C was alsoused in the test bench circuit 200 of FIG. 2 with Z_(i) formed of acapacitance C_(i)=1.0 μF and resistance R_(o)=1.0 kΩ, and the values ofthe components in FIG. 1C selected as C₂=1.0 μF, R₃=1.0 kΩ, and R_(i) asa variable in the range 1.0 kΩ-50.0 kΩ. The voltage across theresistance R_(o) was monitored, and the results obtained are shown inplot 600 of FIG. 6. Inspection of FIG. 6 shows that the circuit behavesas a bandpass filter with variable Q and center frequency. This confirmsthat the circuit of FIG. 1C emulates a lossy positive floatinginductance. The simulation results show that the center frequencychanges from 22.0 Hz to 151.0 Hz. This is in excellent agreement withthe calculations showing that the center frequency changes between 22.5Hz and 159.0 Hz.

With the values of the components in FIG. 1C selected as C₂=47.0 μF,R₃=25.0 kΩ, and R₁ as a variable in the range 9.17 kΩ-50.0 kΩ, theproposed emulated positive inductor was tested using the test benchcircuit of FIG. 2 with C_(i)=4.7 μF and resistance R_(o)=1.0 kΩ, and thevoltage across the resistance R_(o) was monitored. The results obtainedare shown in plot 700 of FIG. 7. Inspection of FIG. 7 shows that thecircuit behaves as a bandpass filter with variable Q and centerfrequency. The simulation results show that the center frequency changesin the range 0.303 Hz-0.7075 Hz. This is in excellent agreement with thecalculations showing that the center frequency changes in the range0.3030 Hz-0.7076 Hz. This confirms that the circuit of FIG. 1C emulatesa lossy positive floating inductance and can be used in designingbandpass filters with center frequencies in the sub-Hz region.

The proposed lossless floating positive capacitance obtainable from FIG.1C was used in the test bench circuit of FIG. 2 with Z_(i) and Z_(o)formed of resistances R_(i)=R_(o)=1.0 kΩ and the values of thecomponents in FIG. 1C selected as C₁=1.0 μF, R₂=10.0 kΩ, and R₃=1.0 kΩ.The voltage across the emulated capacitor and the current through itwere monitored, and the results are shown in plot 800 of FIG. 8, wherethe current is leading the voltage by 90°. This confirms that thecircuit 100 c of FIG. 1C emulates a lossless positive capacitance. Theproposed circuit 100 c of FIG. 1C was also tested as a capacitancemultiplier by connecting it in the test bench circuit 200 of FIG. 2 withfirst external impedance 202 Z_(i) and Z_(o) (second external impedance204) formed of resistances R_(i)=R_(o)=1.0 kΩ, and the values of thecomponents of circuit 100 c in FIG. 1C selected as C₁=1.0 nF, R₃=5.0 kΩ,and R₂ as a variable in the range 10.0 kΩ-65.0 kΩ. The value of theemulated capacitance was monitored, and the results are shown in plot900 of FIG. 9. Inspection of plot 900 of FIG. 9 clearly shows that withR₂=20.0 kΩ, the circuit 100 c of FIG. 1C emulates a capacitance=4.175nF, while the calculated value is C=4.0 nF, and with R₂=70.0 kΩ, thecircuit emulates a capacitance=13.65 nF, while the calculated value is14.0 nF. Thus, a capacitance multiplication by a factor of 15.0 can beachieved. However, the multiplication factor is not linearly increasingwith the value of R₂ for values of R₂>70.0 kΩ. This may be attributed tothe nonidealities of the CFOAs.

The proposed circuit 100 c of FIG. 1C was also tested as a resistancemultiplier by connecting it in the test bench circuit of FIG. 2 withZ_(i) and Z_(o) formed of resistances R_(i)=R_(o)=1.0 kΩ, and with thevalues of the components of circuit 100 c in FIG. 1C selected as R₁=1.0kΩ, R₂=1.0 kΩ, and R₃ as a variable in the range 1.0 kΩ-500.0 kΩ. Thevalue of the emulated resistance was monitored, and the results areshown in plot 1000 of FIG. 10. Inspection of FIG. 10 clearly shows thata resistance multiplication by a factor of 500 can be achieved. However,the multiplication factor is not linearly increasing with the value ofR₃ for values of R₃>100.0 kΩ. This may be attributed to thenonidealities of the CFOAs.

The proposed circuit of FIG. 1C was also tested as an inductancemultiplier by connecting it in the test bench circuit of FIG. 2 withZ_(i) and Z_(o) formed of resistances R_(i)=R_(o)=1.0 kΩ, and with thevalues of the components in FIG. 1C selected as L₁=1.0 mH, R₂=1.0 kΩ,and R₃ variable in the range 1.0 kΩ-25.0 kΩ. The value of the emulatedinductance was monitored, and the results are shown in plot 1100 of FIG.11. Inspection of FIG. 11 clearly shows that an inductancemultiplication factor by 25.0 can be achieved.

The proposed circuit of FIG. 1C was also tested as a frequency-dependentnegative-resistance by connecting it in the test bench circuit of FIG. 2with R_(i)=R_(o)=20.0 kΩ, and with the values of the components in FIG.1C selected as R₁=1.0 kΩ, L₃=200.0 mH and C₂=1.0 μF. The voltage acrossand the current through the emulated frequency-dependentnegative-resistance were monitored, and the results are shown in plot1200 of FIG. 12. Inspection of FIG. 12 clearly shows that the phaseshift between the current and the voltage is 180°. This is a clearindication that the circuit is emulating a negative resistance.Moreover, at frequency=2 kHz, the calculated negative resistance is31.55 kΩ, which is in excellent agreement with the simulated value of29.956 kΩ.

The proposed negative lossless floating inductor obtainable from FIG. 1Dwas used in the test bench circuit of FIG. 2 with Z_(i) formed of avariable positive inductance (L_(i)) and Z_(o) formed of resistance(R_(o)). With the inductance L_(i) varying in the range 54.5 mH-60.0 mH,and the resistance R_(o)=500Ω, and the values of the components in FIG.1D selected as R₁=1.0 kΩ, R₃=1.0 kΩ and C₂=50 nF to emulate a losslessnegative inductance=−50 mH, the applied input voltage and the resultingcurrent though the circuit 200 of FIG. 2 were monitored. The resultsobtained are shown in plot 1300 of FIG. 13. Inspection of FIG. 13clearly shows that the current through the circuit and the applied inputvoltage are in phase, indicating that the total impedance is purelyresistive only when the externally connected inductance is equal to 54.5mH. Otherwise, the current and the voltage are not in phase, indicatingthat the total impedance is inductive. This confirms that the proposedcircuit of FIG. 1D emulates a negative inductance=−54.5 mH, with anerror equal to 9.5% between the calculated and simulated values.

A catalog comprising four new circuits for emulating immittancefunctions has been presented. Each circuit uses three CFOAs and threepassive elements. Using a test bench circuit, the proposed circuits weretested for realizing bandpass filters; capacitance, resistance, andinductance multipliers; and cancellation of positive inductances andresistances. The results show that bandpass filters working in thesub-Hz region are obtainable, and cancellation of positive inductancesand resistances is feasible. The simulation results obtained are inexcellent agreement with the calculations. The maximum error obtained is9.5% for the case of emulating a negative inductance. For large valuesof a multiplying factor, in the case of resistance, capacitance andinductance multipliers, the multiplying factor exhibits a slightnonlinearity. This is attributed to the nonideal characteristics of theCFOAs. Moreover, the proposed emulators do not require any matchingconditions.

It is to be understood that the present invention is not limited to theembodiments described above, but encompasses any and all embodimentswithin the scope of the following claims.

We claim:
 1. A floating immittance emulator circuit, comprising: a firstcurrent feedback operational amplifier (CFOA) having first y-, x-, z-,and w-terminals; a second CFOA having second y-, x-, z-, andw-terminals, the second z-terminal of the second CFOA being connected toground, and the second y-terminal of the second CFOA being connected tothe first z-terminal of the first CFOA; a third CFOA having third y-,x-, z-, and w-terminals, the third z-terminal being connected to thefirst x-terminal of the first CFOA; a first impedance having a firstlead connected to the first y-terminal of the first CFOA and a secondlead connected to the third x-terminal of the third CFOA; a secondimpedance having a first lead connected to the third w-terminal of thethird CFOA and a second lead connected to the third y-terminal of thethird CFOA; and a third impedance having a first lead connected to thethird y-terminal of the third CFOA and a second lead connected to thesecond x-terminal of the second CFOA; wherein the emulator circuitemulates immittance between a voltage v₁ applied to the first y-terminalof the first CFOA and a voltage v₂ applied to the second y-terminal ofthe second CFOA.
 2. The floating immittance emulator according to claim1, wherein the first, second and third impedances comprise a resistanceR₁, a capacitive reactance, $\frac{1}{{sC}_{2}},$ and a resistance R₃,respectively, the emulator simulating a lossy inductance with aseries-connected resistance with R_(eq)=R₁, L_(eq)=C₂R₁R₃.
 3. Thefloating immittance emulator according to claim 2, further comprising: afirst external impedance connected in series with an input of theimmittance emulator; and a second external impedance connected betweenan output of the immittance emulator and ground; wherein a circuitformed by the first and second external impedances in combination withthe immittance emulator defines a transfer function characterizing abandpass filter having a variable Q and center frequency.